9.3 Conversion to Monge form

To compute the angles that the tangents to the lines of curvature at the umbilics make with the axes, the surface has to be set in Monge form for each umbilic separately. Therefore for each umbilic on the surface, a coordinate transformation is needed. But before we conduct the transformation, we need to locate all umbilical points. The principal curvature functions are defined in (3.49), (3.50) as . If we set

then umbilic occurs precisely at a point where the function is zero. Since is a real valued function, it follows that . Consequently, an umbilic occurs where the function has a global minimum.

If the surface representation is non-degenerate and smooth, then , and hence are smooth. Although we are particularly interested in Bézier surfaces which are , we relax the continuity assumption for by assuming that is at least smooth which is guaranteed if the surface is . Already the assumption of differentiability for , which is weaker than , and the condition that has a global minimum at the umbilic implies that at an umbilic. Therefore the governing equation for locating the umbilics are given by

If the surface is a polynomial parametric surface patch (e.g. a Bézier patch), then we denote where and are polynomials in , . Hence (9.26) reduce to an overconstrained system of nonlinear polynomial equations (see also Sect. 8.2.3)

A robust and efficient solution technique based on the interval projected polyhedron algorithm to solve a system of nonlinear polynomial equations is discussed in Chap. 4.

Consider a global frame - and a surface with an umbilical point as illustrated in Fig. 9.3. The umbilical point is represented by a position vector given by:

Figure 9.3: Definition of coordinate system

To represent the surface in the Monge form at the umbilic , we need to attach an orthogonal Cartesian reference frame to it, say - , and we represent a surface point in the frame - as . We choose unit vectors , , as directions of , and axes as shown in Fig. 9.3, where is the tangential vector in direction and is the unit normal vector of the surface at the umbilic.

If we concatenate these three unit vectors , , in a single matrix, we obtain a description of the orientation of the Monge form with respect to the frame - which is called a rotation matrix

Then the relation between and is:

Using (9.30), we can solve for as a function of that is the coordinate of expressed in frame - as a function of the coordinate of point expressed in - frame as

where is the inverse matrix of . Since is an orthonormal matrix, can be replaced by the transpose matrix , therefore

or equivalently

where subscript denotes that the expressions are evaluated at ( , ). In (9.33) through (9.35), is the only term that is the function of and , whereas all terms involving and are evaluated at ( , ). Now we want to express and as functions of and , i.e. and , using (9.33) and (9.34), so that can be written as the same form as (9.2):

According to the inverse function theorem [76,305], this is possible if and only if

If we set and as

then the determinant can be evaluated using the vector identity (3.15) as follows:

Since we are assuming a regular surface, (9.40) will never vanish, and hence we can apply the inverse function theorem. To evaluate in (9.6) we need to compute which can be computed using the chain rule as follows:

The partial derivatives of with respect to and can be obtained easily from (9.35). We can determine , , and by using the inverse function theorem [76,305] as follows:

Hence

We can also evaluate the higher-order derivatives such as , , , , , , , , , , , , , using the chain rule. Once are obtained, we can compute the angles of tangent lines to the lines of curvature passing through the umbilic using (9.10) to (9.20). Since the angles are evaluated in the -plane we need to map back to the parametric -space for integration. Consider a point on the tangent line which passes through the origin and lies on the -plane, say . Then the point can be expressed in terms of using the vectors along the and axes:

Therefore the angle between -axis and the tangent of the line of curvature in the parametric space is given by